MCR3U1
U5L3
The five-point method is a convenient way to sketch the graphs of the sine and cosine functions and their transformations.
One cycle of the sine or cosine function is divided into 5 key points: y = sin  y
1 -
–1 -

90 0

180 0
  
270 0 360 0 y = cos  y
1 -
–1 -

90 0

180 0
  
270 0 360 0
( NOTE: the horizontal axis is divided into four parts!! )

MCR3U1
U5L3
PART A ~ AMPLITUDE CHANGES (VERTICAL STRETCHES/COMPRESSIONS)
The amplitude of a trigonometric function is given by: A = max 𝑣𝑎𝑙𝑢𝑒 − min 𝑣𝑎𝑙𝑢𝑒
2
Ex.
Graph the following functions for one complete cycle. Include a sketch of the parent function, y = sin  or y = cos  , and describe each transformation. a) y = 3sin  y = ½sin  y
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____________________________________________________________
 b) y = –2cos  y = ½cos  y
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____________________________________________________________

AMPLITUDE CHANGES ( y = a sin  )
Transformation Amplitude a a > 1
0 < a < 1 a < 0
NOTE: the amplitude is always positive!!

MCR3U1
PART B ~ PERIOD CHANGES (HORIZONTAL STRETCHES/COMPRESSIONS)
U5L3
The horizontal length of one cycle is called the period . The period of the parent function ( y = sin  or y = cos  ) is 360 0 .
To determine the period of a transformed function: period =
360 0
|𝑘|
Ex.
Graph the following functions for one complete cycle. Include a sketch of the parent function, y = sin  or y = cos  , and describe each transformation. a) y = sin 2  period = y = sin ½  period =
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_________________________________________________________ y b) y = cos 3  period = y

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
PERIOD CHANGES ( y = sin k  )
Transformation Period k k > 1
0 < k < 1 k < 0
NOTE: the value of k tells us how many waves occur in 360 0 !!

MCR3U1
U5L3
Write an equation and graph one cycle for each of the following. Include a sketch of the parent function and state the range. a) sine function amplitude = 3 period = 180 0 y
Range: ________________________________________ b) cosine function amplitude = 2.5 period = 900 0 y
Range: ________________________________________
HOMEWORK: p.379 #1acd, 2bcef



MCR3U1
U5L3
1. State the range for each of the following: a) y = 5sin x b) y = –6.5cos x
2. State the period for each of the following: a) y = cos 6 x b) y = sin ¼ x
3. Sketch one cycle of the graph of each of the following: a) y = 3sin x b) y = –1.5cos x
4. Write the equation for each function described below: c) e) a) c) y y
= sin 3 x
= –½sin ½ x sine function amplitude 6 period 120 0 sine function
ANSWERS: 1.a) R = { y  R / –5  y  5 }
2.a) 60 0
4.a) y = 6sin 3 x
c) y = 4sin x d) y = cos
¼ x f) b) d) y = 4cos 2 x cosine function amplitude 0.5 period 720 b) 1440 0
0 cosine function b) R = { y  R / –6.5  y  6.5 } b) y = 0.5cos 0.5
x d) y =
5
2
cos
3
5 x –
1
2
(or y = 2.5cos 0.6
x – 0.5)